Composable Core-sets for Determinant Maximization: A Simple Near-Optimal Algorithm
This work addresses a computational bottleneck in large-scale data optimization for applications like diversity and fairness, offering incremental improvements over existing methods.
The paper tackles the problem of constructing composable core-sets for determinant maximization, which relates to diversity and fairness modeling, by providing a theoretical guarantee for the Greedy algorithm and proposing a Local Search algorithm that achieves a nearly optimal approximation bound of O(k)^{2k}.
``Composable core-sets'' are an efficient framework for solving optimization problems in massive data models. In this work, we consider efficient construction of composable core-sets for the determinant maximization problem. This can also be cast as the MAP inference task for determinantal point processes, that have recently gained a lot of interest for modeling diversity and fairness. The problem was recently studied in [IMOR'18], where they designed composable core-sets with the optimal approximation bound of $\tilde O(k)^k$. On the other hand, the more practical Greedy algorithm has been previously used in similar contexts. In this work, first we provide a theoretical approximation guarantee of $O(C^{k^2})$ for the Greedy algorithm in the context of composable core-sets; Further, we propose to use a Local Search based algorithm that while being still practical, achieves a nearly optimal approximation bound of $O(k)^{2k}$; Finally, we implement all three algorithms and show the effectiveness of our proposed algorithm on standard data sets.